Comp 521 – Files and Databases Fall 2010 1

Relational Algebra

Study Chapter 4.1-4.2

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Relational Query Languages

  Query languages: Allow manipulation and retrieval of data from a database.

  Relational model supports simple, powerful QLs:   Strong formal foundation based on logic.   Allows for much optimization.

  Query Languages != programming languages!   QLs not expected to be “Turing complete”.   QLs not intended to be used for complex calculations.   QLs support easy, efficient access to large data sets.

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Formal Relational Query Languages

  Two formal Query Languages are the basis for “real” query languages (e.g. SQL):   Relational Algebra: Operational, it provides a

recipe for evaluating the query. Useful for representing execution plans.

  Relational Calculus: Lets users describe what they want, rather than how to compute it. (Non-operational, declarative.)

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Preliminaries

  A query is applied to relation instances, and the result of a query is also a relation instance.   Schemas of input relations for a query are fixed (but

queries are meaningful regardless of instance!)   The schema for the result of a given query is also

fixed! Determined by definition of query language constructs.

  Positional vs. named-field notation:   Positional notation (i.e. R[0]) easier for formalism,

named-field notation (i.e. R.name) more readable.   Both available in SQL

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What is an “Algebra”   Set of operands and operations that they are

“closed” under all compositions   Examples

  Boolean algebra - operands are the logical values True and False, and operations include AND(), OR(), NOT(), etc.

  Integer algebra - operands are the set of integers, operands include ADD(), SUB(), MUL(), NEG(), etc. many of which have special in-fix operator symbols (+,-,*,-)

  In our case “operands” are relations, what are the operators?

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Example Instances

  “Sailors” and “Reserves” relations for our examples.

  We’ll use positional or named field notation, assume that names of fields in query results are “inherited” from names of fields in query input relations.

R1

S1

S2

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Relational Algebra

  Basic operations:   Selection ( ) Selects a subset of rows from relation.   Projection ( ) Deletes unwanted columns from relation.   Cross-product ( ) Allows us to combine two relations.   Set-difference ( ) Tuples in reln. 1, but not in reln. 2.   Union ( ) Tuples in reln. 1 and in reln. 2.

  Additional operations:   Intersection, join, division, renaming: Not essential, but

(very!) useful.

  Since each operation returns a relation, operations can be composed! (Algebra is “closed”.)

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Projection

  Deletes attributes that are not in projection list.

  Schema of result contains exactly the fields in the projection list, with the same names that they had in the (only) input relation.

  Projection operator has to eliminate duplicates! (Why??)   Note: real systems typically

don’t do duplicate elimination unless the user explicitly asks for it. (Why not?)

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Selection

  Selects rows that satisfy selection condition.

  No duplicates in result! (Why?)

  Schema of result identical to schema of (only) input relation.

  Result relation can be the input for another relational algebra operation! (Operator composition.)

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σrating>8(S2)

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Union, Intersection, Set-Difference

  All of these operations take two input relations, which must be union-compatible:   Same number of fields.   ‘Corresponding’ fields

have the same type.   What is the schema of result?

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Cross-Product   Each row of S1 is paired with each row of R1.   Result schema has one field per field of S1 and R1,

with field names `inherited’ if possible.   Conflict: Both S1 and R1 have a field called sid.

  Renaming operator:

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Joins   Condition Join:

  Result schema same as that of cross-product.   Fewer tuples than cross-product, might be

able to compute more efficiently   Sometimes called a theta-join.

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Joins

  Equi-Join: A special case of condition join where the condition c contains only equalities.

  Result schema similar to cross-product, but only one copy of fields for which equality is specified.

  Natural Join: Equijoin on all common fields.

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Division

  Not supported as a primitive operator, but useful for expressing queries like:

Find sailors who have reserved all boats.   Let A have 2 fields, x and y; B have only field y:

  A/B =   i.e., A/B contains all x tuples (sailors) such that for every y

tuple (boat) in B, there is an xy tuple in A.   Or: If the set of y values (boats) associated with an x value

(sailor) in A contains all y values in B, the x value is in A/B.

  In general, x and y can be any lists of fields; y is the list of fields in B, and x y is the list of fields of A.

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Examples of Division A/B

A

B1 B2

B3

A/B1 A/B2 A/B3

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Expressing A/B Using Basic Operators

  Division is not essential op; just a useful shorthand.   (Also true of joins, but joins are so common that systems

implement joins specially.)

  Idea: For A/B, compute all x values that are not “disqualified” by some y value in B.   x value is disqualified if by attaching y value from B, we

obtain an xy tuple that is not in A.

Disqualified x values:

A/B: all disqualified tuples

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Relational Algebra Examples   Assume the following extended schema:

  Sailors( sid: integer, sname: string, rating: integer, age: real)

  Reserves(sid: integer, bid: integer, day: date)   Boat(bid: integer, bname: string, bcolor: string)

  Objective: Write a relational algebra expression whose result instance satisfies the specified conditions   May not be unique   Some alternatives might be more efficient (in

terms of time and/or space)

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Names of sailors who’ve reserved boat #103

  Solution 1:

  Solution 2:

  Solution 3:

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Names of sailors who’ve reserved a red boat

  Information about boat color only available in Boats; so need an extra join:

  A more efficient solution:

A query optimizer can find this, given the first solution!

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Sailors who’ve reserved a red or a green boat

  Can identify all red or green boats, then find sailors who’ve reserved one of these boats:

  Can also define Tempboats using union! (How?)

  What happens if is replaced by in this query?

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Sailors who’ve reserved a red and a green boat

  Previous approach won’t work! Must identify sailors who’ve reserved red boats, sailors who’ve reserved green boats, then find the intersection (note that sid is a key for Sailors):

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Names of sailors who’ve reserved all boats

  Uses division; schemas of the input relations to / must be carefully chosen:

  To find sailors who’ve reserved all ‘Interlake’ boats:

.....

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Summary   The relational model has rigorously defined

query languages that are simple and powerful.

  Relational algebra is more operational; useful as internal representation for query evaluation plans.

  Several ways of expressing a given query; a query optimizer should choose the most efficient version.